On solving the chlorine transport model via Laplace transform

This paper analyzes the two-dimensional chlorine-transport model in pipes. The studied model is in the form of a second-order partial differential equation with a set of boundary conditions. Obtaining exact solution for the current model is a challenge due to the nature of the involved boundary conditions, especially, when applying the Laplace transform. However, such difficulties are solved via implementing the method of residues. The exact solution is obtained in terms of the Bessel functions. The expression for a dimensionless cup-mixing average concentration is also derived analytically. The proposed approach is validated via numerical examples for comparing the results with those in the literature. The present analysis/approach is effective/straightforward and can be further applied on other similar models under different boundary conditions.

The study of the water quality is a growing field of research due to its importance in industry and engineering sciences. As a result of its importance in the quality of drinking water, chlorine is used in most parts of the world as a guarantee for the distribution of safe drinking water 1,2 . To ensure the quality of the water, part of the chlorine must remain to prevent the survival and growth of any microorganisms that may be present during the movement of water in the networks. Therefore, maintaining a certain level of chlorine ensures that no harmful by-products are formed within the distribution networks. In Ref. 3 , the percentage of chlorine concentration that ensures the avoidance of public health risks has been determined as 0.2 mg l −1 . Accordingly, the successful management of ensuring the quality of drinking water requires adherence to the aforementioned chlorine concentration and ensuring that this percentage does not rise above the specified limit. The basic model of chlorine transport was established by Biswas et al. 4 in the form: and where u(x, r) is the chlorine concentration, and A 0 , A 1 and A 2 are dimensionless parameters related to the chlorine decay. Further, details about the dimensionless parameters and derivation of the Eqs. (1)-(4) can be found in Ref. 4 . The purpose of the present work is to solve the system (1)-(4) through applying the Laplace transform (LT). The LT is a well-known and effective approach to solving various scientific models in physics and engineering [5][6][7][8][9][10] . Ebaid and Al sharif 6 applied the LT on the ODE governing the heat transfer of nanofluids suspended with carbon-nanotubes. Ebaid et. al 7 provided the analytic solution for a class of singular boundary value problems (SBVPs) via the LT. Khaled 8 obtained the exact solution of the model describing the radiation  [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] . In this paper we consider the application of the LT for the solution of the system (1)- (4). The paper is structured as follows.
The LT approach is applied in "Application of LT" on Eqs. (1)-(4). Section "Analysis and exact solution of the chlorine decay model" is devoted to obtaining exact solution. Section "Discussion of results" analyses the results and discusses their physical meaning. In addition, the results are to be compared with those in Ref. 4 . Finally, Section "Conclusion" outlines the main conclusions.

Application of LT
Applying LT on Eq. (1) with respect to the variable x, we can write which gives where L(·) and s denote the Laplace operator and variable, respectively. After introducing the BC given in (2), Eq. (6) becomes Re-arranging Eq. (7), yields which is the well known Bessel differential equation with inhomogeneous part (− 1 A 0 ) . The solution of Eq. (8) is given by where J 0 (·) and Y 0 (·) are Bessel functions and c 1 and c 2 denote unknown constants, i = √ −1 . Since u(x, r) and its LT, U(s, r), must be bounded at r = 0 , the value of c 2 must be zero since Y 0 i s+A 1 A 0 r → ∞ as r → 0 . Thus, we can write and where J ′ 0 ( r) = − J 1 ( r) . From Eqs. (10) and (11), we obtain Applying LT on (4) yields From Eqs. (12) and (13), we obtain Substituting (14) into (10) leads to .
Exact solution of the chlorine decay model. The main challenge of this paper is to obtain the inverse LT of the expression in Eq. (20). The expression (20) is really complex due to the nature of the boundary conditions (2)(3)(4). This is because the denominator in expression (20) involves Bessel functions of first and second kind which leads to actual difficulties when deriving the inverse LT of the expression in Eq. (20). However, such difficulties are overcome through applying the method of residues as indicated below.
has simple zeros at s = −A 1 and i s+A 1 A 0 = 1 , 2 , . . . n , . . . , and thus we find simple poles at s 1 = −A 1 and s 2 = −A 1 − A 0 2 n , n = 1, 2, 3, . . . . Therefore, the inverse LT of F(s, r), i.e., f(x, r) can be obtained from Theorem 1 by calculating the residues (Res) of e sx F(s, r) at s 1 = −A 1 and s 2 = −A 1 − A 0 2 n , and then by taking their sum. At s 1 = −A 1 , we have The limit of G(s, r) as s → s 2 can be calculated using the L'Hospital's rule as follows where Since the functions J 0 and J 2 are even and J 1 is odd, we obtain From (29) and (31), we get (Res) s 2 = lim lim s→s 2 (G(s, r)) = lim s→s 2 (s − s 2 )

Discussion of results
The dimensionless cup-mixing average concentration is defined by . .
In order to have a numerical comparison between the current calculations of the first three roots 1 , 2 , and 3 of Eq. (36) and the corresponding results in 4 (using Eqs. (46)-(48)), we may write Eq. (36) as a function of in the form: Figures 1, 2, 3, 4 and 5) highlight details about the roots of f ( ) for different values of A 2 . We verify that we have an infinite number of roots. In addition, all the roots of f ( ) , excepting the first one, are nearly identical for small A 2 ∈ (0.01, 1] , as shown by Fig. 1. However, for the range 1 ≤ A 2 < 10 , Fig. 2 reveals that the first seven roots are considerably different, while the others have approximately the same values. Figure 3 indicates that the first two roots are nearly identical, while the rest of roots are considerably different for the range 10 ≤ A 2 < 45 . For higher values of A 2 , namely in the ranges 50 ≤ A 2 < 90 and 100 ≤ A 2 < 900 , Figs. 4 and 5 reveal that the roots of f ( ) are nearly identical. However, the results introduced in 4 were mainly depend on obtaining the first three roots 1 , 2 and 3 and, therefore, the proposed approach is more accurate.    www.nature.com/scientificreports/ Table 1 presents the three roots 1 , 2 and 3 of Eq. (36) and the corresponding results from 4 (using Eq. (46)) in the interval A 2 ∈ (0.01, 1] . The calculations of the present roots are accomplished through MATHEMATICA. The absolute errors listed in Table 1 show that the results presented in 4 agree with the obtained ones only up to two/three digits at most. This means that the new approach leads to better results than those of 4 after 3 decimal places. A similar conclusion is also obtained in Table 2 regarding 1 , 2 and 3 for the range 1 ≤ A 2 < 10. Table 3 shows that the absolute errors increase in the range 10 ≤ A 2 < 1000 . Such differences in the values may lead to differences when calculating the chlorine concentration or the cup-mixing average concentration. The behavior of the cup-mixing average concentration u av , at the outlet x = 1 of a pipe, versus A 1 are displayed in Figs. 6, 7, 8 and 9 for several values of A 0 and A 2 . These figures indicate that the u av is always a decreasing function in the parameter A 1 . This means that the cup-mixing average concentration decays with increasing the parameter A 1 . In conclusion, the proposed approach gives a clear and precise solution of the mathematical model.   www.nature.com/scientificreports/

Conclusion
The two-dimensional chlorine-model was theoretically solved. The exact solution was obtained by applying the LT. The method of residues was adopted to obtain the inverse LT of complex expressions and the solution was expressed in terms of Bessel functions of the first and the second kinds of order zero. The exact solutions agree with those published previously using the method of separation of variables. However, the obtained numerical results are superior to than those reported in 4 due to the limitations on the calculation of the roots of Eq. (36). Previous studies conducted the numerical calculations based on various fitting curves to predict the values of such roots at prescribed values of A 2 . Such fitting scheme involves some numerical errors as verified in Tables  (1-3), where the absolute errors may be relevant for many cases. The expression of the dimensionless cup-mixing average concentration was analytically derived. The results proved that the new approach gives reliable and accurate solutions of the problem.